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338                                APPENDIX A
(c)      sin (x ± y) = sin x cos ?/ + cos x sin ( =fc ?/) .
(d)      cos (x =fc ?/) = cos a; cos ?/ — sin .r sin ( =fc •?/).
/N      ,     /    ,    \      tan x + tan (dby/)
(e)      tan (x =b ?/) = : - - - ; - _^ .
1 — tan x tan (± ;//)
(f ) sin 2 a; = 2 sin x cos x. (g) cos 2 a; = cos2£ — sin 2 tan x
. in2 a;.   F(f), (g), and (h) are obtained by le'
LtinS y = J5 iu W, W), and (c). -
(i)      sin a; = 2 sin- cos™»
Phcsc may "      '"2
(j)     cos x = cos2 - — sin2 ™-      [These may be obtained by repla< (k)
x                    ng        :> m (f)? (fi), and (h).
I tan -
(1)      sin2 s - i (1 - cos 2 s).   rThege may bc obtainwl wwily from fe. (m)    cos2a; = HI + cos 2j?). [_
x     1 (h)     sin2- = ™ (1— cos a;).     FTlicac are obtained by replacing x b
2t          £J
*% in (I) and (in).
(0)                                             °
.     fTl
?:
.    L2
Angle between two lines.
(p)     cos 6 = cos a cos a' + cos 0 cos $' + cos 7 cos 7'.
V. MACLAURIN'S   THEOREM.
VI. IMPORTANT FUNCTIONS EXPRESSED AS POWKK SKRIt The following expansions are curried out by Maclaurin'H
/IN
(b)
= 1 + x.   [When x^ I, and o«nH(Hiu<»ntly *?'\ f*> etc.,
i   i to     x*     w:3  ,  .r4   ,
44 o;.   [When x<C 1, and consequently •*"% ^*3, etc.,